#358184

order	512
nilpotency class	3
isoclinism	isoclinic class of #23439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321621, #72321838
maximal normal subgroup of	#72321621, #72321838

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265380);