#493717

order	512
nilpotency class	4
isoclinism	isoclinic class of #480274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1179 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,d^c=di,f^c=fi,g^c=ghi,g^f=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, 400913);