#196555

order	512
nilpotency class	3
isoclinism	isoclinic class of #162486
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1158 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)(5,6,9,11,10,14,8,13)(7,12,15,18,16,20,17,19)(21,22)(23,29,31,36)(24,28)(25,27)(26,30,34,32)(33,35);(6,11,14,13)(12,18,20,19)(21,23,24,32,25,31,33,30)(22,26,28,36,27,34,35,29);(2,4)(5,7,10,16)(6,12,14,20)(8,17,9,15)(11,18,13,19);(5,8,10,9)(6,13,14,11)(7,15,16,17)(12,18,20,19)(22,27)(26,34)(28,35)(29,36)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=h,e^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,g^b=gi,d^c=dh

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, 103751);