#7551008

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	C₂×#3280	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2134 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=g,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

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, 7458204);