#29656

order	256
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	Aut(G) order 49152
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁴
prime radical	C₂³×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

714 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 714 conjugacy classes of subgroups — too many to draw. Only its 121 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) = 24
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23);(5,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24);(12,20)(15,22)(18,24)(21,23)

Polycyclic presentation:

2^8|a^2=e,b^2=e,b^a=be,c^a=cf,g^a=gh,c^b=cg,f^b=fh,d^c=de

Conjugacy classes & character table

37 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#29656⋊C₃, #72227695, #72227726, …
maximal subgroup of	#245248, #245250, #245253, …
maximal normal subgroup of	#245248, #245250, #245253, …
maximal quotient of	#327771, #327772, #327773, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 22644);