#373866

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1274 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281062);