#1080498

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2764 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fghi,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	#72326557, #72326607, #72326625
maximal normal subgroup of	#72326557, #72326607, #72326625
maximal quotient of	#72328856

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 987694);