#5954433

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
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	#15258	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2999 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fhi,c^2=ghi,d^2=fhi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328079, #72328138, #72328222
maximal normal subgroup of	#72328079, #72328138, #72328222
maximal quotient of	#72329043

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5861629);