#6629390

order	512
nilpotency class	3
isoclinism	isoclinic class of #32898
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4496 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|b^2=i,e^2=fi,b^a=bf,c^a=cg,d^a=dh,h^c=hi,g^d=gi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72245692, #72246847, #72246879, …
maximal subgroup of	#72231727, #72236596, #72236649, …
maximal normal subgroup of	#72231727, #72236596, #72236649, …
maximal quotient of	#72231703, #72249970, #72250212, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536586);