#408621

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

1320 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)(13,14,19,25)(15,26,28,20)(16,31,34,39)(17,24,18,23)(21,33,22,32)(27,40,41,44)(29,37,30,38)(35,42,36,43);(1,2)(3,4)(13,15,18,30,19,28,17,29)(14,20,24,38,25,26,23,37)(16,27,33,43,34,41,32,42)(21,35,39,40,22,36,31,44);(2,4)(5,7,9,8)(6,10,11,12)(13,16,18,33,19,34,17,32)(14,21,24,39,25,22,23,31)(15,27,30,43,28,41,29,42)(20,35,38,40,26,36,37,44);(5,8,9,7)(6,10,11,12)(13,16,17,32,19,34,18,33)(14,22,23,39,25,21,24,31)(15,27,29,42,28,41,30,43)(20,36,37,40,26,35,38,44)

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,d^c=di

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, 315817);