#195948

order	512
nilpotency class	3
isoclinism	isoclinic class of #184767
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1358 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ci,e^b=eh,g^b=gh,d^c=dh

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, 103144);