#1409579

order	512
nilpotency class	2
isoclinism	isoclinic class of #1393169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#14885	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3015 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)(7,12)(8,10)(9,11)(14,16)(18,20)(23,28)(24,26);(5,7,9,8)(6,10,11,12)(13,14,17,16)(15,18,19,20)(21,22,25,27)(23,26,28,24);(7,8)(10,12)(13,15)(14,18)(16,20)(17,19)(21,23)(22,26)(24,27)(25,28);(1,2,3,4)(5,8,9,7)(6,10,11,12)(13,16,17,14)(15,18,19,20);(5,6)(7,10)(8,12)(9,11)(13,15)(14,18)(16,20)(17,19)(21,24,25,26)(22,23,27,28)

Polycyclic presentation:

2^9|b^2=fg,d^2=hi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=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, 1316775);