#415054

order	512
nilpotency class	3
isoclinism	isoclinic class of #394189
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1124 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=e,d^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^c=fi

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, 322250);