#7688954

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234910, #72234925, #72234926, …
maximal normal subgroup of	#72234910, #72234925, #72234926, …
maximal quotient of	#72261146, #72261566

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7596150);