#188498

order	512
nilpotency class	3
isoclinism	isoclinic class of #160780
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1166 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ehi,c^2=h,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,f^c=fi,g^c=ghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95694);