#183855

order	512
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄²	C₂⁴
commutator, socle	C₂³	C₂²×C₄²
prime radical	C₂²×D₄	C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

830 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 366 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=f,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=dhi,c^b=ch,d^b=dhi

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91051);