#941198

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=gh,c^2=ghi,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 848394);