#346949

order	512
nilpotency class	3
isoclinism	isoclinic class of #346934
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2148 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 2148 conjugacy classes of subgroups — too many to draw. Only its 190 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,e^b=eh,f^b=fi,e^c=ei,f^c=fi,g^d=gh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72272611
maximal subgroup of	#72272568, #72272611
maximal normal subgroup of	#72272568, #72272611
maximal quotient of	#72340883, #72341050

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254145);