#4435253

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 2097152
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

1952 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)(7,11)(9,12)(15,19)(18,20)(21,22)(23,25)(24,31)(26,27)(28,33)(29,35)(30,32)(34,36);(5,6,8,10)(7,9,11,12)(13,14,17,16)(15,18,19,20)(22,27)(23,29)(28,31)(30,36);(5,6)(7,9)(8,10)(11,12)(13,15)(14,18)(16,20)(17,19)(21,23,26,29)(22,25,27,35)(24,30,33,36)(28,34,31,32);(1,2,3,4)(13,16,17,14)(15,20,19,18)(21,24,26,33)(22,28,27,31)(23,30,29,36)(25,34,35,32);(5,7,8,11)(6,9,10,12)(21,25)(22,29)(23,27)(24,32)(26,35)(28,30)(31,36)(33,34)

Polycyclic presentation:

2^9|b^2=fgh,c^2=f,d^2=gi,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, 4342449);