#508105

order	512
nilpotency class	4
isoclinism	isoclinic class of #508104
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#29636
commutator, Frattini, cosocle	C₂×#41	C₂⁴
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

851 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 851 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415301);