#7182118

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

1752 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)(15,23)(16,20)(17,31)(18,21)(19,27)(22,32)(24,39)(25,34)(26,38)(28,42)(29,37)(30,44)(33,40)(35,41)(36,43);(5,6,9,10)(7,8,12,11)(13,15,18,26)(14,19,21,34)(16,28,29,24)(17,25,32,27)(20,35,37,33)(22,38,31,23)(30,40,43,41)(36,39,44,42);(2,4)(5,7)(6,11)(8,10)(9,12)(13,16,18,29)(14,20,21,37)(15,24,26,28)(17,30,32,43)(19,33,34,35)(22,36,31,44)(23,39,38,42)(25,41,27,40);(6,10)(7,12)(14,21)(15,25)(16,29)(17,32)(19,23)(24,40)(26,27)(28,41)(33,42)(34,38)(35,39)(36,44);(5,8,9,11)(6,7,10,12)(14,21)(15,26)(16,29)(22,31)(25,27)(30,43)(33,35)(39,42)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7089314);