#512690

order	512
nilpotency class	5
isoclinism	isoclinic class of #2263
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₁₆
commutator	C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂²	C₂²×D₁₆
prime radical	C₂²×#508	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

784 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2,3,4)(5,6)(7,18)(8,14)(9,27)(10,31)(11,13)(12,20)(15,29)(16,25)(17,26)(19,46)(21,51)(22,53)(23,45)(24,54)(28,43)(30,59)(32,41)(33,40)(34,50)(35,47)(36,61)(37,63)(38,58)(39,56)(42,67)(44,49)(48,65)(52,68)(55,64)(57,60)(62,66);(5,7)(6,12)(8,19)(9,28)(10,32)(11,23)(13,34)(14,35)(15,41)(16,43)(17,44)(18,27)(20,29)(21,25)(22,26)(24,55)(30,62)(31,36)(33,48)(37,65)(38,66)(39,51)(40,64)(42,68)(45,59)(46,54)(47,56)(49,67)(50,63)(52,58)(53,60)(57,61);(6,13)(12,34)(14,17)(15,37)(16,38)(18,45)(27,59)(31,40)(35,44)(36,64)(39,42)(41,65)(43,66)(46,53)(51,68)(54,60);(5,8,11,26)(6,14,13,17)(7,19,23,22)(9,24,30,57)(10,25,33,58)(12,35,34,44)(15,39,37,42)(16,40,38,31)(18,46,45,53)(20,47,50,49)(21,48,52,32)(27,54,59,60)(28,55,62,61)(29,56,63,67)(36,43,64,66)(41,51,65,68)

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=gh,g^2=hi,h^2=i,b^a=be,c^a=ci,e^a=eg,g^a=gh,h^a=hi,e^b=eg,g^b=gh,h^b=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#33981, #72322680, #72322858, …
maximal normal subgroup of	C₄×#33981, #72322680, #72322858, …
maximal quotient of	#72322606, #72322759

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419886);