#188038

order	512
nilpotency class	3
isoclinism	isoclinic class of #188035
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×#548	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

3618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,g^b=gh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95234);