#188447

order	512
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1174 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)(6,10,11,12)(13,14)(15,21,23,19)(16,20)(17,18)(22,27,28,26)(24,25);(1,2,3,4)(14,18)(19,21)(20,25)(26,27);(5,6)(7,10)(8,12)(9,11)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27);(5,7,9,8)(6,10,11,12)(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95643);