#7255306

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	C₂×#447	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1544 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,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, 7162502);