#246808

order	512
nilpotency class	3
isoclinism	isoclinic class of #246799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#45977	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1335 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1335 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) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^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,d^b=dh,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, 154004);