#188748

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
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×C₄⋊C₄
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1052 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(6,10)(8,15)(11,16)(12,17)(14,19)(18,20)(21,22,25,28,26,29,24,34)(23,30,32,35,33,36,31,27);(5,6,8,10,9,12,15,17)(7,11,14,16,13,18,19,20)(21,23,24,31,26,33,25,32)(22,27,28,30,29,35,34,36);(2,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20);(6,12)(7,13)(10,17)(14,19)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95944);