#182792

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1076 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)(5,6,9,12)(7,15,16,20)(8,14,10,13)(11,18,19,17)(21,22,26,30)(23,31,32,36)(24,29,25,28)(27,34,35,33);(5,7,10,18,9,16,8,17)(6,11,14,20,12,19,13,15)(21,23)(22,27)(24,33)(25,34)(26,32)(28,31)(29,36)(30,35);(1,3,4,2)(7,16)(11,19)(15,20)(17,18)(23,32)(27,35)(31,36)(33,34);(6,12)(11,19)(13,14)(15,20)(21,24,26,25)(22,28,30,29)(23,33,32,34)(27,31,35,36)

Polycyclic presentation:

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

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, 89988);