#861527

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

2774 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:

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

Polycyclic presentation:

2^9|b^2=fh,c^2=fgi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326672, #72326716, #72326738
maximal normal subgroup of	#72326672, #72326716, #72326738
maximal quotient of	#72328811

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 768723);