#7533694

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#45316	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1999 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)(7,11)(8,10)(9,12)(13,14,19,23)(15,24,28,41)(16,21,31,39)(17,22,18,33)(20,26,35,27)(25,40,30,44)(29,36,32,38)(34,43,37,42);(1,2)(3,4)(7,12)(9,11)(13,15,17,26,19,28,18,27)(14,20,33,41,23,35,22,24)(16,25,32,43,31,30,29,42)(21,34,36,44,39,37,38,40);(5,7,8,12)(6,9,10,11)(13,16)(14,21)(15,25)(17,32)(18,29)(19,31)(20,34)(22,38)(23,39)(24,40)(26,43)(27,42)(28,30)(33,36)(35,37)(41,44);(2,4)(6,10)(9,11)(13,17,19,18)(14,22,23,33)(15,26,28,27)(16,29,31,32)(20,24,35,41)(21,36,39,38)(25,42,30,43)(34,44,37,40);(13,15,17,26,19,28,18,27)(14,20,33,41,23,35,22,24)(16,30,32,42,31,25,29,43)(21,37,36,40,39,34,38,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,d^2=i,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei

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, 7440890);