#7283694

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
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	#23871	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1658 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)(17,18,23,25,22,28,21,27)(19,24,35,37,31,40,30,39)(20,32,36,42,33,26,34,41)(29,43,46,48,44,38,45,47);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,31)(18,24,28,40)(20,29,33,44)(21,30,23,35)(25,37,27,39)(26,38,32,43)(34,45,36,46)(41,47,42,48);(10,13)(11,15)(18,25)(21,23)(24,37)(26,41)(27,28)(30,35)(32,42)(34,36)(38,47)(39,40)(43,48)(45,46);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,20,22,33)(18,26,28,32)(19,29,31,44)(21,34,23,36)(24,38,40,43)(25,41,27,42)(30,45,35,46)(37,47,39,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(20,33)(26,32)(29,44)(34,36)(38,43)(41,42)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,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, 7190890);