#4464132

order	512
nilpotency class	2
isoclinism	isoclinic class of #4462490
automorphism	Aut(G) order 25165824
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂³×C₄⁴

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

3728 conjugacy classes of subgroups; only the 554 normal subgroups are shown.

normal subgroups only

This group has 3728 conjugacy classes of subgroups — too many to draw. Only its 554 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|d^2=gi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,d^c=df

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4371328);