#392196

order	512
nilpotency class	3
isoclinism	isoclinic class of #32947
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3392 conjugacy classes of subgroups; only the 498 normal subgroups are shown.

normal subgroups only

This group has 3392 conjugacy classes of subgroups — too many to draw. Only its 498 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)(6,11)(9,12)(13,21)(14,22)(15,23)(16,29)(17,24)(19,25)(26,35)(27,32)(28,33)(31,36);(5,6,12,23,10,15,9,11)(7,13,24,32,18,27,17,21)(8,14,25,33,20,28,19,22)(16,26,34,35,29,36,30,31);(5,7)(6,13)(8,16)(9,17)(10,18)(11,21)(12,24)(14,26)(15,27)(19,30)(20,29)(22,31)(23,32)(25,34)(28,36)(33,35);(5,8)(6,14)(7,16)(9,19)(10,20)(11,22)(12,25)(13,26)(15,28)(17,30)(18,29)(21,31)(23,33)(24,34)(27,36)(32,35)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299392);