#7182188

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1745 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:

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

Polycyclic presentation:

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

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, 7089384);