#154073

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
commutator	C₂³	C₂²×C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2970 conjugacy classes of subgroups; only the 914 normal subgroups are shown.

normal subgroups only

This group has 2970 conjugacy classes of subgroups — too many to draw. Only its 914 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) = 512

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=e,b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=de

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 61269);