#265291

order	512
nilpotency class	3
isoclinism	isoclinic class of #265283
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62245	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 999 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) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=i,d^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gi,c^b=cg,f^b=fi,f^c=fh

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, 172487);