#102914

order	512
nilpotency class	3
isoclinism	isoclinic class of #102905
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

1029 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1029 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,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, #72267798
maximal normal subgroup of	#72217725, #72267798
maximal quotient of	#72216703, #72267800, #72267808

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10110);