#619683

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3872 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331992, #72331994, #72332060
maximal normal subgroup of	#72331992, #72331994, #72332060
maximal quotient of	#72333189

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 526879);