#350453

order	512
nilpotency class	3
isoclinism	isoclinic class of #346346
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23930	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1316 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 1316 conjugacy classes of subgroups — too many to draw. Only its 168 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,24)(20,30)(21,32)(22,27)(23,28)(25,31)(26,29);(2,5)(6,8)(10,14)(15,16)(17,19,23,22)(18,24,28,27)(20,29,21,31)(25,30,26,32);(1,3,4,7)(2,6,5,8)(10,15)(12,13)(14,16)(17,20,23,21)(18,25,28,26)(19,29,22,31)(24,30,27,32);(10,16)(12,13)(14,15)(17,21,23,20)(18,26,28,25)(19,29,22,31)(24,30,27,32)

Polycyclic presentation:

2^9|b^2=hi,c^2=eh,d^2=hi,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gh,d^b=dhi,f^c=fh,g^c=gi,f^d=fi,g^d=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264245, #72264620, #72264639
maximal normal subgroup of	#72264245, #72264620, #72264639

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 257649);