#431458

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=ef,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 338654);