#355912

order	512
nilpotency class	3
isoclinism	isoclinic class of #3198
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,17,23,19,24,18,22)(15,20,27,36,26,34,28,35)(16,29,30,21,32,37,31,38)(25,39,41,33,40,44,42,43);(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,25)(17,27)(18,28)(19,26)(21,33)(22,35)(23,36)(24,34)(29,39)(30,41)(31,42)(32,40)(37,44)(38,43);(2,4)(7,12)(9,11)(13,16,18,31,19,32,17,30)(14,21,23,37,24,38,22,29)(15,25,28,42,26,40,27,41)(20,33,36,44,34,43,35,39);(6,10)(9,11)(15,26)(20,34)(25,40)(27,28)(33,43)(35,36)(39,44)(41,42)

Polycyclic presentation:

2^9|a^2=g,c^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,h^a=hi,c^b=ce,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267920
maximal normal subgroup of	#72267920
maximal quotient of	#72267923, #72267927

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263108);