#498062

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

594 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=ehi,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^b=de,e^b=eh,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319320, #72319486, #72319560
maximal normal subgroup of	#72319320, #72319486, #72319560

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405258);