#495827

order	512
nilpotency class	4
isoclinism	isoclinic class of #33517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	#33579	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

684 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6,10,14,11,16,27,15)(7,17,20,32,22,36,21,29)(8,13,23,33,24,34,9,25)(12,28,30,18,31,26,35,19);(1,2,3,4);(5,7)(6,12)(8,19)(9,26)(10,20)(11,22)(13,32)(14,30)(15,35)(16,31)(17,25)(18,24)(21,27)(23,28)(29,34)(33,36);(5,8,10,23,11,24,27,9)(6,13,14,33,16,34,15,25)(7,18,20,26,22,19,21,28)(12,29,30,17,31,32,35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#2179, #72319557, #72319561
maximal normal subgroup of	D₄×#2179, #72319557, #72319561

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403023);