#188392

order	512
nilpotency class	3
isoclinism	isoclinic class of #29592
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,c^b=ci,f^c=fh

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, 95588);