#414029

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
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

1236 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)(16,31,35,39)(17,36,29,21)(18,25,19,24)(22,33,40,34)(28,37,43,44)(32,41,42,38);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,17,20,30,18,29)(14,21,25,23,26,36,24,27)(16,28,34,32,35,43,33,42)(22,37,31,38,40,44,39,41);(6,9)(10,12)(13,16)(14,22)(15,28)(17,32)(18,33)(19,34)(20,35)(21,37)(23,38)(24,39)(25,31)(26,40)(27,41)(29,42)(30,43)(36,44);(5,7,8,11)(6,10,9,12)(13,17,19,30,20,29,18,15)(14,23,25,36,26,27,24,21)(16,32,34,43,35,42,33,28)(22,38,31,44,40,41,39,37)

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,d^b=di,f^c=fi

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, 321225);