#500303

order	512
nilpotency class	4
isoclinism	isoclinic class of #499949
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3176	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

816 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(7,17)(8,22)(9,10)(12,26)(14,30)(15,16)(18,35)(19,33)(20,34)(21,23)(24,25)(27,36)(28,31)(29,32);(1,2)(3,4)(5,6)(7,18,19,28)(8,14)(9,15)(10,16)(11,13)(12,20,27,21)(17,31,33,35)(22,30)(23,36,34,26)(24,29)(25,32);(2,4)(5,7,10,20,11,19,9,21)(6,12,16,18,13,27,15,28)(8,23,22,33,24,34,25,17)(14,31,30,36,29,35,32,26);(6,13)(8,24)(12,27)(15,16)(17,33)(18,28)(22,25)(23,34)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407499);