#3021902

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2822 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=fi,d^2=fg,e^2=i,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, 2929098);