#7308845

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1444 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,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=dfhi,e^b=eh,d^c=dh,e^c=ei,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, 7216041);