#146703

order	512
nilpotency class	4
isoclinism	isoclinic class of #12707
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1047 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1047 conjugacy classes of subgroups — too many to draw. Only its 190 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 — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,d^b=di,f^b=fhi,e^c=ei,f^c=fhi,f^d=fi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53899);