#408885

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

1106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

(5,6)(7,12)(8,11)(9,10)(13,14,20,15,18,23,19,26)(16,24,28,22,17,25,27,21);(1,2)(3,4)(6,10)(11,12)(13,15,19,23,18,26,20,14)(16,24,27,22,17,25,28,21);(2,4)(5,7,9,8)(6,11,10,12)(13,16,19,27,18,17,20,28)(14,21,15,24,23,22,26,25);(5,8,9,7)(6,11,10,12)(13,17,19,28,18,16,20,27)(14,22,15,25,23,21,26,24)

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=f,d^2=f,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,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, 316081);