#4423072

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 4194304
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2818 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,c^2=f,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

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, 4330268);