#8093108

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

13852 conjugacy classes of subgroups; only the 2938 normal subgroups are shown.

normal subgroups only

This group has 13852 conjugacy classes of subgroups — too many to draw. Only its 2938 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) = 24
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 6 generators on 24 points:

(3,7)(6,8)(10,13)(14,16)(17,18,21,23)(19,20,24,22);(2,5)(6,8)(11,15)(14,16)(17,19,21,24)(18,22,23,20);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16);(9,11)(10,14)(12,15)(13,16)(17,20)(18,19)(21,22)(23,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,20)(21,24)(22,23);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,19)(21,22)(23,24)

Polycyclic presentation:

2^9|a^2=g,b^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72227705, #72227811, C₂×#72209711, …
Fitting subgroup of	#72227610, #72227700, #72237118, …
maximal subgroup of	#72209710, #72209711, #72209767, …
maximal normal subgroup of	#72209710, #72209711, #72209767, …
maximal quotient of	#72209699, #72210056, #72210057, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000304);