#254262

order	512
nilpotency class	3
isoclinism	isoclinic class of #254227
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

802 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 136 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=i,d^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=deh,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,g^b=gi,d^c=di,e^c=eh,f^c=fi,g^c=ghi,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161458);