#102907

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	C₂×#2011	C₂
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

1229 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|c^2=g,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	#72217708, #72217722, #72217734
maximal normal subgroup of	#72217708, #72217722, #72217734
maximal quotient of	#72215007, #72217563

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10103);