#331328

order	512
nilpotency class	3
isoclinism	isoclinic class of #331095
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2867	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 964 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=h,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238524);