#494996

order	512
nilpotency class	4
isoclinism	isoclinic class of #33691
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

586 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=ehi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,h^a=hi,f^b=fi,e^c=ei,f^c=fh,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250810, #72252775, #72252778, …
maximal normal subgroup of	#72250810, #72252775, #72252778, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402192);