#60049

order	256
nilpotency class	2
isoclinism	isoclinic class of #58990
automorphism	Aut(G) order 196608
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁵
prime radical	C₂⁴	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 817 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) = 128

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

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

Polycyclic presentation:

2^8|a^2=f,b^2=fg,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	#72340888
Fitting subgroup of	#60049⋊C₃
maximal subgroup of	#97516, #172079, #172080, …
maximal normal subgroup of	#97516, #172079, #172080, …
maximal quotient of	#532346, #532990, #534154, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53037);