#606804

order	512
nilpotency class	2
isoclinism	isoclinic class of #53158
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2690 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)(7,11)(8,9)(10,12)(14,15)(16,20)(22,25)(26,28);(6,8)(10,11)(14,15)(18,19)(23,27)(26,28);(13,14,17,15)(16,18,20,19)(21,22)(23,26)(24,25)(27,28);(5,7,9,12)(6,10,8,11)(13,15,17,14)(16,19,20,18)(21,23,24,27)(22,26,25,28);(5,8,9,6)(7,11,12,10)(13,16)(14,18)(15,19)(17,20)(21,22)(23,26)(24,25)(27,28)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dfh,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 514000);