#520964

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

2510 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)(7,9)(8,11)(10,12)(13,14)(15,20)(16,19)(17,18)(21,22)(23,27)(24,25)(26,28);(5,7)(6,8)(9,12)(10,11)(21,22)(23,26)(24,25)(27,28);(5,8,10,9)(6,11,12,7)(14,18)(19,20)(22,25)(26,27);(6,12)(7,11)(13,15,17,16)(14,19,18,20)(21,23,24,28)(22,26,25,27);(5,9,10,8)(6,7,12,11)(13,16,17,15)(14,19,18,20)

Polycyclic presentation:

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

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, 428160);