#392031

order	512
nilpotency class	3
isoclinism	isoclinic class of #32873
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2840
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1622 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1622 conjugacy classes of subgroups — too many to draw. Only its 234 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235766, #72235771, #72235788, …
maximal normal subgroup of	#72235766, #72235771, #72235788, …
maximal quotient of	#72235809

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299227);