#408618

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	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1356 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 315814);