#6372715

order	512
nilpotency class	3
isoclinism	isoclinic class of #60388
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4650 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318111, #72318196, #72318466, …
maximal normal subgroup of	#72318111, #72318196, #72318466, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279911);