#2564232

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15257	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2949 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327832, #72327872, #72327915
maximal normal subgroup of	#72327832, #72327872, #72327915
maximal quotient of	#72328501

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2471428);