#508151

order	512
nilpotency class	4
isoclinism	isoclinic class of #508078
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#3487	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

830 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 124 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,g^f=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415347);