#196949

order	512
nilpotency class	3
isoclinism	isoclinic class of #162342
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1300 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 104145);