#4466350

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
automorphism	Aut(G) order 1048576
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3689 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263602, #72263614, #72263618
maximal normal subgroup of	#72263602, #72263614, #72263618

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4373546);