#535274

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

2220 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=fghi,e^2=hi,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, 442470);