#238067

order	512
nilpotency class	3
isoclinism	isoclinic class of #238065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

919 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 919 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,6)(4,7)(5,8)(9,14)(10,13)(11,15)(12,16)(17,18)(19,29)(20,34,38,44)(21,28)(22,27)(23,26)(24,32)(25,37,42,36)(30,46,48,41)(31,40)(33,43)(35,45,47,39);(1,3)(2,6)(4,9)(5,10)(7,14)(8,13)(11,16)(12,15)(17,19,22,32)(18,24,27,29)(20,35)(21,33,23,31)(25,41)(26,43,28,40)(30,37)(34,45)(36,48)(38,47)(39,44)(42,46);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(17,20)(18,25)(19,30)(21,36)(22,38)(23,37)(24,39)(26,34)(27,42)(28,44)(29,45)(31,47)(32,48)(33,35)(40,46)(41,43);(4,11)(7,15)(9,16)(12,14)(17,21,22,23)(18,26,27,28)(19,31,32,33)(20,36,38,37)(24,40,29,43)(25,34,42,44)(30,47,48,35)(39,46,45,41)

Polycyclic presentation:

2^9|a^2=eh,b^2=e,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,d^b=dh,f^b=fi,d^c=di,e^c=eh,f^c=fh,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145263);