#416720

order	512
nilpotency class	3
isoclinism	isoclinic class of #394267
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1808 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1808 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=eh,c^2=ef,d^2=hi,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=fi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323916);