#431536

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

842 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)(3,4)(5,6,11,15,10,14,23,9)(7,16,20,26,19,12,28,18)(8,21,24,30,25,31,27,13)(17,33,34,29,22,36,35,32);(5,7,10,19)(6,12,14,16)(8,22)(9,26,15,18)(11,20,23,28)(13,29)(17,25)(21,36)(24,35)(27,34)(30,32)(31,33);(5,8,23,24,10,25,11,27)(6,13,9,21,14,30,15,31)(7,17,28,34,19,22,20,35)(12,29,26,36,16,32,18,33);(2,4)(5,9,23,14,10,15,11,6)(7,18,28,12,19,26,20,16)(8,21,24,30,25,31,27,13)(17,33,34,29,22,36,35,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 338732);