#345629

order	512
nilpotency class	3
isoclinism	isoclinic class of #333407
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

572 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 252825);