#359647

order	512
nilpotency class	3
isoclinism	isoclinic class of #23882
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3485	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3902 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260846, #72261267, #72261310, …
maximal normal subgroup of	#72260846, #72261267, #72261310, …
maximal quotient of	#72260901, #72302324, #72314876

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266843);