#4125359

order	512
nilpotency class	2
isoclinism	isoclinic class of #4122169
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

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

2994 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)(5,6)(7,9)(8,11)(10,12)(13,14)(15,19)(16,17)(18,20)(21,22,24,27)(23,26,28,25);(6,7)(10,11)(14,17)(15,20)(21,23,24,28)(22,25,27,26);(8,12)(10,11)(14,17)(18,19)(21,22,24,27)(23,26,28,25);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,15,16,20)(14,18,17,19)(21,23)(22,26)(24,28)(25,27);(5,8)(6,10)(7,11)(9,12)(13,14)(15,19)(16,17)(18,20)(21,22)(23,25)(24,27)(26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4032555);