#520561

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

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

2370 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 427757);