#512545

order	512
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₆ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂⁴
commutator	C₂	C₂²×C₄×C₁₆
socle	C₂²	C₂³×C₁₆
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,e^2=g,g^2=h,h^2=i,b^a=bi,d^c=di

Conjugacy classes & character table

272 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231854, #72231859, #72231861, …
maximal normal subgroup of	#72231854, #72231859, #72231861, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419741);