#493846

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1152 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)(5,6,9,16,10,17,25,35,11,14,26,31,8,15,23,32)(7,18,21,33,22,13,29,30,19,12,24,27,20,28,36,34);(5,7)(6,12)(8,20)(9,24)(10,22)(11,19)(13,15)(14,18)(16,33)(17,28)(21,26)(23,29)(25,36)(27,31)(30,35)(32,34);(2,4)(5,7)(6,13,14,28)(8,20)(9,24)(10,22)(11,19)(12,15,18,17)(16,34,31,30)(21,26)(23,29)(25,36)(27,35,33,32);(7,19)(12,18)(13,28)(20,22)(21,24)(27,33)(29,36)(30,34)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401042);