#499475

order	512
nilpotency class	4
isoclinism	isoclinic class of #499263
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3150	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

587 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 587 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,g^b=gh,h^b=hi,d^c=dh,e^c=ei,g^c=gh,h^c=hi,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322650
maximal normal subgroup of	#72322650

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406671);