#20612

order	256
nilpotency class	3
isoclinism	isoclinic class of #3026
automorphism	Aut(G) order 49152
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#120
commutator	C₂³	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂	C₂×#379
prime radical	C₂×#585	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1820 conjugacy classes of subgroups; only the 301 normal subgroups are shown.

normal subgroups only

This group has 1820 conjugacy classes of subgroups — too many to draw. Only its 301 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:

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

Polycyclic presentation:

2^8|a^2=g,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^b=fh,g^b=gh,e^c=eh

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#20612⋊C₃, #72236738, #72236783, …
maximal subgroup of	#147753, #147754, #147757, …
maximal normal subgroup of	#147753, #147754, #147757, …
maximal quotient of	#184905, #184906, #184909, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13600);