#494053

order	512
nilpotency class	4
isoclinism	isoclinic class of #33688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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×#575	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

955 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)(7,15)(8,11)(9,22)(10,25)(12,17)(13,24)(14,21)(16,30)(18,34)(19,31)(20,35)(23,28)(26,33)(27,36)(29,32);(6,11)(9,20)(12,26)(14,25)(17,33)(23,32)(24,36)(28,29);(5,7)(6,12)(8,19)(9,23)(10,16)(11,26)(13,30)(14,28)(15,22)(17,24)(18,21)(20,32)(25,29)(27,34)(31,35)(33,36);(7,16,19,18)(9,20)(12,28,26,29)(13,27)(15,30,31,34)(17,23,33,32)(22,35)(24,36)

Polycyclic presentation:

2^9|a^2=g,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,f^b=fi,d^c=dh,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	#72239799, #72239803, #72239840, …
maximal normal subgroup of	#72239799, #72239803, #72239840, …
maximal quotient of	#72265529, #72265707

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401249);