#246784

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
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	C₂×C₄²	C₂⁴
prime radical	#45978	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1174 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1174 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,10,14)(7,15,17,20)(8,13,9,12)(11,18,19,16)(21,22,26,30)(23,31,24,33)(25,35,36,29)(27,34,28,32);(1,2)(3,4)(5,7)(6,11)(8,18)(9,16)(10,17)(12,20)(13,15)(14,19)(21,23,26,24)(22,27,30,28)(25,34,36,32)(29,33,35,31);(2,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,15)(13,20)(14,19)(25,36)(29,35)(31,33)(32,34);(5,7)(6,11)(8,18)(9,16)(10,17)(12,20)(13,15)(14,19)(21,24,26,23)(22,28,30,27)(25,32,36,34)(29,31,35,33)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,c^b=cg,e^b=eh,f^b=fi,e^c=ei,f^c=fhi,e^d=eh,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153980);