#244125

order	512
nilpotency class	3
isoclinism	isoclinic class of #232766
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2397
prime radical	C₂³	#525
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

584 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151321);