#7149304

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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×#3221	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1904 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)(7,11)(10,12)(13,14)(15,23)(16,29)(17,33)(18,21)(19,28)(20,31)(22,24)(25,41)(26,36)(27,35)(30,39)(32,38)(34,43)(37,42)(40,44);(6,9)(7,11)(13,15,24,26,18,27,17,28)(14,19,33,35,21,36,22,23)(16,30,40,42,31,25,32,43)(20,37,44,39,29,34,38,41);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,25)(17,32)(18,31)(19,34)(21,29)(22,38)(23,39)(24,40)(26,43)(27,30)(28,42)(33,44)(35,41)(36,37);(6,9)(10,12)(15,26)(16,31)(17,24)(19,35)(20,29)(22,33)(23,36)(25,42)(27,28)(30,43)(34,39)(37,41);(5,7,8,11)(6,10,9,12)(14,21)(15,27)(16,31)(22,33)(26,28)(32,40)(34,37)(39,41)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7056500);