#7224223

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1643 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fi,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7131419);