#7502966

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1845 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=dfi,e^b=ei,g^c=gi,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7410162);