#6432221

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431707
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2434 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,h^b=hi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308854, #72308882
maximal normal subgroup of	#72308854, #72308882

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339417);