#4928851

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

5734 conjugacy classes of subgroups; only the 998 normal subgroups are shown.

normal subgroups only

This group has 5734 conjugacy classes of subgroups — too many to draw. Only its 998 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=fh,c^2=gh,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	#72266416
Fitting subgroup of	#72268760, #72268833
maximal subgroup of	#72266414, #72266415, #72268760, …
maximal normal subgroup of	#72266414, #72266415, #72268760, …
maximal quotient of	#72266422

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4836047);