#219901

order	512
nilpotency class	3
isoclinism	isoclinic class of #164512
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

1148 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)(7,10)(8,15,16,11)(9,12)(13,19,20,17)(14,18)(21,22)(23,31)(24,30)(25,27)(26,29)(28,32)(33,36)(34,35);(1,2)(3,4)(5,7,9,14)(6,10,12,18)(8,13,16,20)(11,17,15,19)(22,27)(28,35)(29,30)(31,36);(5,8,9,16)(6,11,12,15)(7,13,14,20)(10,17,18,19)(21,23,25,33)(22,28,27,35)(24,32,26,34)(29,31,30,36);(2,4)(21,24,25,26)(22,29,27,30)(23,32,33,34)(28,31,35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,d^b=dei,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, 127097);