#7145924

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	#2397	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2161 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=f,e^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=eh,d^c=dh,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, 7053120);