#7502750

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1819 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)(13,14,18,23)(15,25,17,20)(16,28,30,40)(19,34,33,24)(21,35,38,31)(22,32,39,27)(26,37,29,44)(36,42,41,43);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,20)(16,26)(17,18)(19,27)(21,36)(22,24)(23,25)(28,44)(29,30)(31,42)(32,33)(34,39)(35,43)(37,40)(38,41);(2,4)(13,16,18,30)(14,21,23,38)(15,26,17,29)(19,35,33,31)(20,36,25,41)(22,37,39,44)(24,40,34,28)(27,43,32,42);(6,11)(7,12)(13,17,18,15)(14,22,23,39)(16,29,30,26)(19,27,33,32)(20,34,25,24)(21,37,38,44)(28,41,40,36)(31,42,35,43);(5,8,9,10)(6,7,11,12)(15,17)(20,25)(22,39)(26,29)(27,32)(36,41)(37,44)(42,43)

Polycyclic presentation:

2^9|a^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=df,e^b=ei,g^c=gi,e^d=ei,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, 7409946);