#989840

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2407	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3430 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

(2,4)(9,10,14,12)(11,16,15,13)(17,18,21,22)(19,23,24,20);(5,6,7,8)(9,11)(10,13)(12,16)(14,15)(19,24)(20,23);(5,6)(7,8)(9,10,14,12)(11,16,15,13)(17,19)(18,20)(21,24)(22,23);(1,2)(3,4)(9,12,14,10)(11,16,15,13)(17,20)(18,19)(21,23)(22,24);(9,13,14,16)(10,15,12,11)

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,c^2=f,d^2=f,e^2=f,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 897036);