#117624

order	512
nilpotency class	3
isoclinism	isoclinic class of #117624
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1438 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 1438 conjugacy classes of subgroups — too many to draw. Only its 130 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)(3,5)(4,6)(7,8)(9,10)(11,19)(12,23,26,16)(13,27)(14,18)(15,21)(17,29)(20,36,24,34)(22,35)(25,32,28,33)(30,38,40,37)(31,39);(1,3)(2,5)(4,7)(6,8)(9,11,13,21,14,22,29,39)(10,15,17,19,18,31,27,35)(12,24,25,38,26,20,28,37)(16,30,32,36,23,40,33,34);(5,8)(9,12)(10,16)(11,20)(13,28)(14,26)(15,30)(17,33)(18,23)(19,34)(21,38)(22,24)(25,29)(27,32)(31,40)(35,36)(37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,e^b=ei,f^b=fh,d^c=dh,g^c=gh

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, 24820);