#994791

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15124	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3060 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=fh,d^2=gh,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, 901987);