#4453259

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328044, #72328055, #72328069
maximal normal subgroup of	#72328044, #72328055, #72328069
maximal quotient of	#72331409

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4360455);