#2923122

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2682	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3386 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=gh,d^2=fi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2830318);