#3999734

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

2989 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=i,d^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326572, #72326596, #72326619
maximal normal subgroup of	#72326572, #72326596, #72326619
maximal quotient of	#72329129

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3906930);