#299234

order	512
nilpotency class	3
isoclinism	isoclinic class of #299233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60797	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

926 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 926 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,f^b=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, 206430);