#1361343

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

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

3565 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)(7,11)(10,12)(13,14,17,19)(15,20,16,18)(22,26)(23,24);(5,6,9,8)(7,10,11,12)(13,15,17,16)(14,18,19,20)(21,22)(23,28)(24,27)(25,26);(1,2)(3,4)(13,15,17,16)(14,18,19,20)(21,23,25,24)(22,27,26,28);(5,7)(6,10)(8,12)(9,11)(13,16,17,15)(14,18,19,20)(21,24,25,23)(22,27,26,28);(1,2)(3,4)(5,8,9,6)(7,12,11,10)(13,14)(15,18)(16,20)(17,19)(21,22)(23,27)(24,28)(25,26)

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=f,d^2=f,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327581, #72327601, #72327614
maximal normal subgroup of	#72327581, #72327601, #72327614

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1268539);