#536314

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2180 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)(5,6,8,11)(7,10,12,9)(14,18)(15,20);(1,3,4,2)(5,7)(6,9)(8,12)(10,11)(22,25)(26,28);(7,12)(9,10)(13,14,17,18)(15,19,20,16)(21,22,24,25)(23,26,27,28);(5,7)(6,10)(8,12)(9,11)(13,15)(14,19)(16,18)(17,20)(21,23,24,27)(22,26,25,28);(5,6)(7,10)(8,11)(9,12)(13,16,17,19)(14,15,18,20)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fgh,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=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, 443510);