#7501762

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23422	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2457 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)(7,9)(8,10)(11,12)(13,14,17,19)(15,23,25,38)(16,27,28,37)(18,31,30,22)(20,32,34,26)(21,29,35,33)(24,40,41,36)(39,42,44,43);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(14,19)(16,28)(20,34)(22,31)(23,38)(24,41)(29,33)(42,43);(2,4)(5,8,9,12)(6,10,7,11)(13,15,17,25)(14,20,19,34)(16,24,28,41)(18,32,30,26)(21,36,35,40)(22,38,31,23)(27,44,37,39)(29,43,33,42);(6,7)(10,11)(13,16,17,28)(14,21,19,35)(15,24,25,41)(18,33,30,29)(20,36,34,40)(22,27,31,37)(23,39,38,44)(26,43,32,42);(15,25)(20,34)(23,38)(24,41)(26,32)(36,40)(39,44)(42,43)

Polycyclic presentation:

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

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, 7408958);