#126530

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₆ · Schur multiplier (H₂) = C₂×C₄×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈²	C₂²
commutator	C₂	C₄²×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×C₄×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

426 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,h^2=i,c^b=cg

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of	#72234283, #72234284, #72234291
maximal subgroup of	#72210645, #72234226, #72234227, …
maximal normal subgroup of	#72210645, #72234226, #72234227, …
maximal quotient of	#72210619, #72210680, #72234228, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33726);