#528800

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2480 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)(7,11)(8,10)(9,12)(13,14,16,17)(15,19,20,18)(21,22)(23,28)(24,26)(25,27);(5,6,8,10)(7,9,12,11)(14,17)(18,19)(22,24)(27,28);(1,2,4,3)(13,14)(15,18)(16,17)(19,20)(21,23,26,25)(22,27,24,28);(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,17)(19,20)(21,24,26,22)(23,28,25,27);(1,3,4,2)(5,7,8,12)(6,9,10,11)(13,15)(14,18)(16,20)(17,19)(21,25,26,23)(22,28,24,27)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=gi,d^2=gh,e^2=fi,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 435996);