#1129229

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
prime radical	C₂×#2540	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2924 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)(7,10)(8,9)(11,12)(13,14,17,20)(15,16,19,18)(22,24)(23,25);(2,4)(6,9)(10,12)(13,15,17,19)(14,18,20,16)(21,22,26,24)(23,27,25,28);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,19,20,15)(21,23,26,25)(22,27,24,28);(5,7)(6,10)(8,11)(9,12)(13,16)(14,15)(17,18)(19,20)(21,24,26,22)(23,27,25,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,19)(18,20)(21,25,26,23)(22,27,24,28)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=f,d^2=g,e^2=g,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	#72281893, #72281968, #72282061
maximal normal subgroup of	#72281893, #72281968, #72282061

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1036425);