#7560270

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#45265	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1850 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:

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

Polycyclic presentation:

2^9|c^2=fgi,d^2=gi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7467466);