#6459740

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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	C₂×#3054	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3100 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gh,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,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, 6366936);