#528838

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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

2080 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)(5,6,8,9)(7,11,12,10)(13,14,17,19)(15,16,20,18)(21,22,25,24)(23,26,27,28);(6,9)(10,11)(13,15,17,20)(14,18,19,16)(23,27)(26,28);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,20)(18,19)(21,23,25,27)(22,26,24,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,20)(18,19)(21,22,25,24)(23,26,27,28);(1,3,4,2)(5,7,8,12)(6,10,9,11)(13,16)(14,15)(17,18)(19,20)(21,24,25,22)(23,28,27,26)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fi,d^2=fh,e^2=fgi,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, 436034);