#2076487

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
automorphism	?
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	#15267	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3492 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300761, #72300773, #72300790, …
maximal normal subgroup of	#72300761, #72300773, #72300790, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1983683);