#4607072

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3258 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)(7,11)(8,12)(9,10)(13,14,16,18)(15,19,20,17)(21,22,25,28)(23,24,27,26);(6,10)(7,12)(13,14)(15,17)(16,18)(19,20)(21,23,25,27)(22,26,28,24);(6,10)(8,11)(15,20)(17,19)(23,27)(24,26);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,14)(15,17)(16,18)(19,20)(21,22)(23,26)(24,27)(25,28);(5,8)(6,7)(9,11)(10,12)(13,15,16,20)(14,17,18,19)(21,24,25,26)(22,27,28,23)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336634, #72336664, #72336721
maximal normal subgroup of	#72336634, #72336664, #72336721

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4514268);