#188214

order	512
nilpotency class	3
isoclinism	isoclinic class of #188176
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2632 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)(7,13,15,10)(8,11)(9,12)(14,19,20,17)(16,18)(21,22,25,30,26,29,24,28)(23,31,32,36,33,27,34,35);(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23,26,33)(22,27,29,31)(24,32,25,34)(28,35,30,36);(2,4)(8,16)(11,18)(14,20)(17,19)(21,24,26,25)(22,28,29,30)(23,32,33,34)(27,35,31,36);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95410);