#7054127

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2093 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6961323);