#1076544

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂³×C₄³

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

3176 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^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	#72336315, #72336381, #72336449, …
maximal normal subgroup of	#72336315, #72336381, #72336449, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 983740);