#789606

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2750 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)(13,14,16,18)(15,19,20,17)(23,27)(25,28);(5,6,9,7)(8,11,12,10)(13,14,16,18)(15,17,20,19)(21,22,24,26)(23,25,27,28);(1,2)(3,4)(13,14)(15,17)(16,18)(19,20);(5,7,9,6)(8,10,12,11)(13,15)(14,17)(16,20)(18,19)(21,23)(22,25)(24,27)(26,28);(5,8,9,12)(6,10,7,11)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 696802);