#3022072

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1832 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fgh,c^2=gi,d^2=g,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, 2929268);