#184830

order	512
nilpotency class	3
isoclinism	isoclinic class of #184811
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×#121	C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2610 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)(7,15,18,11)(8,12)(9,13)(10,14)(16,20,17,19)(21,22,24,28,25,27,33,35)(23,29,31,36,30,26,32,34);(5,7)(6,11)(8,16)(9,17)(10,18)(12,19)(13,20)(14,15)(21,23)(22,26)(24,32)(25,30)(27,29)(28,36)(31,33)(34,35);(2,4)(5,8,10,9)(6,12,14,13)(7,16,18,17)(11,19,15,20);(5,9,10,8)(6,13,14,12)(7,17,18,16)(11,20,15,19)(23,30)(26,29)(31,32)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92026);