#111518

order	512
nilpotency class	3
isoclinism	isoclinic class of #97080
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂⁵	C₂²×C₄
Frattini, prime radical, cosocle	C₂⁶	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁶ ▷ C₂² ▷ C₁

1850 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1850 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 64

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268310, #72268315, #72268317
maximal normal subgroup of	#72268310, #72268315, #72268317

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18714);