#7208292

order	512
nilpotency class	3
isoclinism	isoclinic class of #7207866
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#53065	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,27,28,40)(20,32,33,24)(21,35,29,25)(23,36,37,31)(30,44,42,41)(34,39,43,47)(38,45,46,48);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(19,28)(20,33)(23,37)(24,32)(27,40)(31,36)(34,43)(39,47);(11,15)(13,16)(17,19,29,36,22,28,21,31)(18,23,35,40,26,37,25,27)(20,30,43,48,33,42,34,45)(24,38,47,44,32,46,39,41);(3,8)(5,7)(10,14)(11,15)(17,20,22,33)(18,24,26,32)(19,30,28,42)(21,34,29,43)(23,38,37,46)(25,39,35,47)(27,41,40,44)(31,45,36,48);(2,6)(5,7)(11,15)(13,16)(20,33)(24,32)(30,42)(34,43)(38,46)(39,47)(41,44)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7115488);