#1075196

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂²×C₄⁴

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

3218 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=h,c^2=h,d^2=gh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303512, #72303529, #72306597
maximal normal subgroup of	#72303512, #72303529, #72306597

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 982392);