#993855

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2424	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,d^2=h,e^2=fg,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, 901051);