#358784

order	512
nilpotency class	3
isoclinism	isoclinic class of #23444
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,b^a=be,c^a=cf,d^a=di,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265876, #72265878, #72265879, …
maximal normal subgroup of	#72265876, #72265878, #72265879, …
maximal quotient of	#72266111, #72302892

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265980);