#946767

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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×#127	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2762 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)(5,6)(7,11)(8,9)(10,12)(14,18)(16,19)(21,22)(23,26)(24,25)(27,28);(5,6,8,9)(7,10,12,11)(13,14,17,18)(15,19,20,16)(21,23,25,28)(22,26,24,27);(6,9)(10,11)(13,15,17,20)(14,16,18,19)(21,23)(22,27)(24,26)(25,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,18)(17,20)(21,24,25,22)(23,26,28,27);(5,7,8,12)(6,10,9,11)(13,16,17,19)(14,15,18,20)

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 853963);