#7697591

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7773 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 20 points:

(1,2)(3,4)(5,6,10,9)(7,12,11,8)(13,14,17,16)(15,19,20,18);(5,7)(6,8)(9,12)(10,11)(13,15,17,20)(14,18,16,19);(2,4)(5,8)(6,11)(7,9)(10,12)(13,14,17,16)(15,18,20,19);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11);(5,9,10,6)(7,12,11,8)(13,16,17,14)(15,19,20,18);(5,7)(6,12)(8,9)(10,11)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234923, #72237366, C₂×#7697591
maximal normal subgroup of	#72234923, #72237366, C₂×#7697591
maximal quotient of	#72261745, C₂×#7697591

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7604787);