#4928936

order	512
nilpotency class	2
isoclinism	isoclinic class of #4928752
automorphism	Aut(G) order 50331648
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄³

Aliases

No aliases for this group.

Subgroups

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

5368 conjugacy classes of subgroups; only the 1002 normal subgroups are shown.

normal subgroups only

This group has 5368 conjugacy classes of subgroups — too many to draw. Only its 1002 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) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of	#72266412
Fitting subgroup of	#72268769, #72268855
maximal subgroup of	#72266410, #72266411, #72268769, …
maximal normal subgroup of	#72266410, #72266411, #72268769, …
maximal quotient of	#72266422

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4836132);