#6371709

order	512
nilpotency class	3
isoclinism	isoclinic class of #60460
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ci,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265535, #72307320, #72308712
maximal normal subgroup of	#72265535, #72307320, #72308712
maximal quotient of	D₄×#1938

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278905);