#221809

order	512
nilpotency class	3
isoclinism	isoclinic class of #164187
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1049 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,d^b=de,f^c=fi,g^c=ghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 129005);