#183248

order	512
nilpotency class	3
isoclinism	isoclinic class of #17603
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1138 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=i,e^2=h,g^2=hi,b^a=be,d^a=dh,e^a=eh,g^a=ghi,e^b=ei,d^c=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90444);