#493751

order	512
nilpotency class	4
isoclinism	isoclinic class of #480263
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1155 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=i,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,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, 400947);