#414026

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
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

1250 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,d^2=gh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,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, 321222);