#7503144

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2796 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7410340);