#105299

order	512
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂³	C₄³
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	C₂×#95
prime radical	C₂⁴	C₂×C₄²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

722 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,e^2=i,f^2=h,b^a=bd,d^a=dh,c^b=ci,e^b=eh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12495);