#221817

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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1178 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)(7,13,15,10)(8,16)(9,12)(11,17)(14,18,20,19)(21,22)(23,27)(24,32,33,36)(25,30)(26,29)(28,31,35,34);(1,2)(3,4)(21,23,26,25)(22,27,29,30)(24,31,33,34)(28,32,35,36);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,24)(22,28)(23,31)(25,34)(26,33)(27,32)(29,35)(30,36);(2,4)(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(22,29)(27,30)(28,35)(32,36)

Polycyclic presentation:

2^9|a^2=g,b^2=i,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, 129013);