#115177

order	512
nilpotency class	3
isoclinism	isoclinic class of #115163
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1208 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1208 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(17,18)(19,25,27,22)(20,28,29,23)(21,24)(26,30)(31,32);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,31,28,30);(2,7)(4,13)(6,9)(10,12)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,g^b=gh,d^c=di,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22373);