#196941

order	512
nilpotency class	3
isoclinism	isoclinic class of #161854
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	C₂³×C₄	C₂⁴
prime radical	#35077	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1539 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,c^b=ch,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, 104137);