#1145463

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
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

3201 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=fghi,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281245, #72281496, #72281908
maximal normal subgroup of	#72281245, #72281496, #72281908

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1052659);