#7627807

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
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
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂³ ▷ C₁

5090 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32);(2,5)(6,8)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32);(9,11,12,16)(10,13,14,15)(19,25)(22,28)(26,31)(29,32);(18,23)(22,28)(24,30)(29,32);(17,20,21,27)(18,24,23,30)(19,26,25,31)(22,29,28,32);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238826
maximal normal subgroup of	#72238826

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7535003);