#7224804

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	#30446	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1625 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=h,e^2=fi,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, 7132000);