#544320

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

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	C₂×#392	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2684 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)(8,10)(11,12)(13,14)(15,19)(16,20)(17,18);(5,7)(6,9)(8,11)(10,12)(14,18)(15,16)(21,22)(23,26)(24,25)(27,28);(6,10)(9,12)(13,15)(14,19)(16,17)(18,20)(22,25)(26,28);(7,11)(9,12)(13,16,17,15)(14,20,18,19);(21,23,24,27)(22,26,25,28)

Polycyclic presentation:

2^9|a^2=i,d^2=f,e^2=fh,b^a=bf,c^a=cg,c^b=ch,d^b=dfg,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, 451516);