#1687399

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

3101 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,5,3)(4,7,8,6)(9,10,13,12)(11,15,16,14)(17,18)(19,24)(20,21)(22,23)(25,26)(27,31)(28,29)(30,32);(2,3)(4,8)(10,12)(14,15)(17,19)(18,21)(20,22)(23,24)(26,29)(27,32);(2,3)(6,7)(11,16)(14,15)(18,23)(19,20)(25,26,28,29)(27,30,32,31);(1,3,5,2)(4,7,8,6)(9,11,13,16)(10,14,12,15)(17,20,22,19)(18,24,23,21)(25,26)(27,30)(28,29)(31,32);(1,4)(2,6)(3,7)(5,8)(9,12,13,10)(11,15,16,14)(17,21,22,24)(18,19,23,20)(25,27,28,32)(26,30,29,31)

Polycyclic presentation:

2^9|a^2=fh,c^2=fghi,d^2=ghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232625, #72232686, #72232701
maximal normal subgroup of	#72232625, #72232686, #72232701

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1594595);