#532207

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁴×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2776 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 439403);