#2975635

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
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
prime radical	#15267	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2763 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336289, #72336403, #72336448
maximal normal subgroup of	#72336289, #72336403, #72336448

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2882831);