#188346

order	512
nilpotency class	3
isoclinism	isoclinic class of #188176
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1206 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)(21,22,25,30,26,29,24,28)(23,31,33,35,34,27,32,36);(1,2,3,4)(5,7)(6,10)(8,13)(9,14)(11,17)(12,18)(15,19)(16,20);(5,8,9,16)(6,11,12,15)(7,13,14,20)(10,17,18,19)(21,23,26,34)(22,27,29,31)(24,33,25,32)(28,36,30,35);(7,14)(10,18)(13,20)(17,19)(21,24,26,25)(22,28,29,30)(23,32,34,33)(27,35,31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95542);