#299535

order	512
nilpotency class	3
isoclinism	isoclinic class of #299505
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1034 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 158 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=f,d^2=ei,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di,g^b=gh,d^c=dhi,g^c=gh

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, 206731);