#544817

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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

2014 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)(6,10)(8,11)(22,25)(26,28);(5,6,9,10)(7,11,12,8)(14,17)(18,20)(23,27)(26,28);(6,10)(7,12)(13,14,16,17)(15,18,19,20)(21,22,24,25)(23,26,27,28);(5,7)(6,11)(8,10)(9,12)(21,23)(22,26)(24,27)(25,28);(5,8,9,11)(6,7,10,12)(13,15,16,19)(14,18,17,20)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fgh,e^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 452013);