#7628135

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂⁴
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂³ ▷ C₁

6678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|e^2=hi,f^2=g,b^a=bg,c^a=ch,c^b=ci,d^b=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233609, #72233690, #72260815, …
maximal normal subgroup of	#72233609, #72233690, #72260815, …
maximal quotient of	#72262085, #72304038, #72304075

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7535331);