#503221

order	512
nilpotency class	4
isoclinism	isoclinic class of #501132
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1160 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=f,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410417);