#188616

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	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1276 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)(5,6,8,10,7,9,11,12)(13,14)(15,22)(16,21)(17,19)(18,25)(20,23)(24,28)(26,27);(1,2,3,4)(6,9)(10,12)(14,19)(20,27)(21,25)(22,28);(6,9)(8,11)(13,15)(14,20)(16,23)(17,24)(18,26)(19,27)(21,28)(22,25);(13,16,17,18)(14,21,19,25)(15,23,24,26)(20,28,27,22)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=i,f^2=i,g^2=hi,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, 95812);