#633915

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
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

4166 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)(6,9)(10,12)(14,17)(15,19)(21,22)(23,27)(24,26)(25,28);(13,14,16,17)(15,18,19,20)(21,22,24,26)(23,25,28,27);(1,2)(3,4)(21,22)(23,25)(24,26)(27,28);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,25)(24,26)(27,28);(5,7,8,11)(6,10,9,12)(13,15,16,19)(14,18,17,20)(21,23,24,28)(22,25,26,27)

Polycyclic presentation:

2^9|a^2=g,b^2=f,e^2=hi,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	#72232708, #72232709, #72233394, …
maximal normal subgroup of	#72232708, #72232709, #72233394, …
maximal quotient of	#72330702, #72330795

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 541111);