#117627

order	512
nilpotency class	3
isoclinism	isoclinic class of #117624
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂⁵	C₂×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1366 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 1366 conjugacy classes of subgroups — too many to draw. Only its 134 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)(9,10,14,18,15,19,13,30)(11,20,24,32,23,36,22,16)(12,25,28,34,29,17,27,33)(21,37,39,40,26,35,38,31);(1,3)(2,5)(4,7)(6,8)(9,11,13,22,15,23,14,24)(10,16,18,20,19,32,30,36)(12,26,27,39,29,21,28,38)(17,31,33,37,25,40,34,35);(5,8)(9,12,15,29)(10,17,19,25)(11,21,23,26)(13,28,14,27)(16,31,32,40)(18,34,30,33)(20,35,36,37)(22,39,24,38)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,g^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, 24823);