#221795

order	512
nilpotency class	3
isoclinism	isoclinic class of #165149
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1102 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)(8,16,18,12)(9,13)(10,14)(15,20,17,19)(21,22)(23,26)(24,27)(25,28)(29,32)(30,34)(31,35)(33,36);(1,2)(3,4)(5,7,10,9)(6,11,14,13)(8,17,18,15)(12,20,16,19)(21,23)(22,26)(24,31)(25,30)(27,35)(28,34)(29,33)(32,36);(5,8)(6,12)(7,15)(9,17)(10,18)(11,19)(13,20)(14,16)(22,27)(23,30)(24,32)(26,33)(28,29)(34,35);(2,4)(5,9,10,7)(6,13,14,11)(8,17,18,15)(12,20,16,19)(22,28)(26,34)(27,29)(33,35)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ch,d^b=de,f^c=fi,g^c=ghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 128991);