#188215

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, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2268 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)(5,6,8,12,9,11,17,19)(7,13,14,20,15,10,16,18)(21,22,25,29,26,30,24,28)(23,27,31,34,32,35,33,36);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18);(2,4)(21,23,26,32)(22,27,30,35)(24,33,25,31)(28,36,29,34);(21,24,26,25)(22,28,30,29)(23,31,32,33)(27,34,35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,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, 95411);