#545111

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, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1998 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=g,e^2=f,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, 452307);