#493725

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×Q₈	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1124 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)(5,6,10,15,11,16,8,21)(7,12,20,28,18,26,17,31)(9,14,22,29,24,30,23,13)(19,27,32,35,33,36,34,25);(1,2,3,4)(5,7)(6,12)(8,17)(9,19)(10,20)(11,18)(13,25)(14,27)(15,28)(16,26)(21,31)(22,32)(23,34)(24,33)(29,35)(30,36);(5,8,11,10)(6,13,16,29)(7,17,18,20)(9,23,24,22)(12,25,26,35)(14,15,30,21)(19,34,33,32)(27,28,36,31);(7,18)(12,26)(17,20)(19,33)(25,35)(27,36)(28,31)(32,34)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=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, 400921);