#115661

order	512
nilpotency class	3
isoclinism	isoclinic class of #115640
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1062 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1062 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=dh,d^c=dh,g^c=gi

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, 22857);