#670054

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2929 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fhi,c^2=i,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 577250);