#138182

order	512
nilpotency class	4
isoclinism	isoclinic class of #137984
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#469	C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

418 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,6,7,5,8,4,3)(9,10)(11,21,24,38)(12,17,27,37)(13,29)(14,20)(15,35)(16,23,28,30)(18,33)(19,34)(22,39)(25,31,40,26)(32,36);(1,3,6,8,5,7,4,2)(9,11,22,37,34,40,32,23,14,24,33,17,15,25,13,30)(10,16,36,26,35,27,39,21,20,28,29,31,19,12,18,38);(1,4,5,6)(2,7,8,3)(9,12,14,27)(10,17,20,37)(11,18,24,39)(13,31,32,26)(15,28,34,16)(19,23,35,30)(21,22,38,33)(25,29,40,36)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=h,d^2=ghi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=ei,f^b=fh,d^c=di,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45378);