#134026

order	512
nilpotency class	4
isoclinism	isoclinic class of #133899
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂³×D₄	C₂³
prime radical	#35463	C₂
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#123 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

1150 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 1150 conjugacy classes of subgroups — too many to draw. Only its 90 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)(3,7,8,5)(4,6)(9,10)(11,19)(12,16)(13,25,28,17)(14,18)(15,21,30,26)(20,36)(22,33)(23,34,27,31)(24,32)(29,37,40,39)(35,38);(9,11)(10,15,18,30)(12,20)(13,26)(14,22)(16,29,32,40)(17,33,25,19)(21,28)(23,39)(24,38)(27,37)(31,35,34,36);(1,3,4,8)(2,5,6,7)(9,12,14,24)(10,16,18,32)(11,20,22,38)(13,27,28,23)(15,29,30,40)(17,34,25,31)(19,35,33,36)(21,39,26,37)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=di,e^b=ei,f^b=fg,d^c=di,f^c=fh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266413, #72268762
maximal normal subgroup of	#72266413, #72268762
maximal quotient of	#72266414, #72266415

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41222);