#102915

order	512
nilpotency class	3
isoclinism	isoclinic class of #102905
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#17068	C₂
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

989 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 989 conjugacy classes of subgroups — too many to draw. Only its 140 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 — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,d^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cf,d^b=dg,e^b=eh,d^c=dh,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217725
maximal normal subgroup of	#72217725
maximal quotient of	#72216328

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10111);