#189489

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1074 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,d^c=di,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 96685);