#7182108

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=i,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,f^b=fi,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, 7089304);