#385531

order	512
nilpotency class	3
isoclinism	isoclinic class of #30437
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2668
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14407
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1087 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ef,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,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, 292727);