#392036

order	512
nilpotency class	3
isoclinism	isoclinic class of #32875
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2842
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1496 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1496 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)(7,17)(8,22)(9,14)(10,15)(11,16)(12,20)(13,25)(18,33)(19,32)(21,28)(23,29)(24,31)(26,30)(27,34)(35,36);(2,4)(5,7,10,21,11,20,9,19)(6,12,14,28,16,17,15,32)(8,23,26,34,25,35,24,18)(13,29,30,33,22,36,31,27);(1,2,3,4)(5,8,9,24,11,25,10,26)(6,13,15,31,16,22,14,30)(7,18,21,23,20,34,19,35)(12,27,28,29,17,33,32,36);(5,9,11,10)(6,14,16,15)(7,19,20,21)(8,24,25,26)(12,28,17,32)(13,30,22,31)(18,23,34,35)(27,36,33,29)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=i,g^2=i,h^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,g^b=gi,d^c=de,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235757, #72235763, #72235790
maximal normal subgroup of	#72235757, #72235763, #72235790
maximal quotient of	#72235809

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299232);