#533956

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, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2182 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=i,e^2=fgh,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, 441152);