#1508276

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2310 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:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=fh,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1415472);