#238459

order	512
nilpotency class	3
isoclinism	isoclinic class of #232313
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#34641	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1408 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145655);