#127504

order	512
nilpotency class	4
isoclinism	isoclinic class of #1339
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#327
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂²	C₂×#327
prime radical	C₂⁵	D₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

854 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gi,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,h^a=hi,d^b=dh,g^b=gi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213046, #72213784, #72214477, …
maximal normal subgroup of	#72213046, #72213784, #72214477, …
maximal quotient of	#72211371

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34700);