#60045

order	256
nilpotency class	2
isoclinism	isoclinic class of #58990
automorphism	Aut(G) order 393216
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

1245 conjugacy classes of subgroups; only the 427 normal subgroups are shown.

normal subgroups only

This group has 1245 conjugacy classes of subgroups — too many to draw. Only its 427 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) = 64

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^8|b^2=f,c^2=fgh,d^2=h,e^2=h,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72344164
Fitting subgroup of	#60045⋊C₃, #72240891, #72340813, …
maximal subgroup of	#97512, #130086, #133446, …
maximal normal subgroup of	#97512, #130086, #133446, …
maximal quotient of	#532345, #532983, #534181, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53033);