#213694

order	512
nilpotency class	3
isoclinism	isoclinic class of #163012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#36624	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1302 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1302 conjugacy classes of subgroups — too many to draw. Only its 222 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)(3,7)(4,6)(5,8)(9,10,12,14,11,13,15,16)(17,18)(19,23)(20,28)(21,25)(22,26)(24,27)(29,32)(30,31);(1,3)(2,5)(4,8)(6,7)(10,13)(14,16)(17,19,22,21)(18,23,26,25)(20,29,30,27)(24,28,32,31);(3,8)(5,7)(10,13)(12,15)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32);(2,6)(5,7)(10,13)(14,16)(17,19,22,21)(18,25,26,23)(20,27,30,29)(24,28,32,31)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ceh,f^c=fi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120890);