#4986980

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

2152 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=fh,d^2=hi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

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, 4894176);