#353974

order	512
nilpotency class	3
isoclinism	isoclinic class of #1913
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂³	C₂²×C₄²
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
socle	C₂²	C₂×#386
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2070 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,4,3)(5,6)(7,15)(8,12)(9,13)(10,14)(11,18)(16,22)(17,28)(19,24)(20,25)(21,26)(23,31)(27,30)(29,33)(32,34)(35,36);(5,7)(6,11,14,15)(8,16)(9,17)(10,18)(12,22,25,27)(13,23,26,28)(19,29)(20,30)(21,31)(24,33,34,35)(32,36);(1,3,4,2)(5,8)(6,12)(7,16)(9,19)(10,20)(11,22)(13,24)(14,25)(15,27)(17,29)(18,30)(21,32)(23,33)(26,34)(28,35)(31,36);(5,9,10,21)(6,13,14,26)(7,17,18,31)(8,19,20,32)(11,23,15,28)(12,24,25,34)(16,29,30,36)(22,33,27,35)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=i,b^a=be,c^a=cf,h^a=hi,e^b=ei,f^b=fi,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314374, #72314402, #72314414
maximal normal subgroup of	#72314374, #72314402, #72314414

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261170);