#7632657

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

8705 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)(5,6,9,8)(7,10,12,11)(14,18)(15,20);(6,8)(10,11)(13,14)(15,19)(16,20)(17,18);(1,2)(3,4)(5,6,9,8)(7,10,12,11)(13,15,17,20)(14,19,18,16);(5,7,9,12)(6,10,8,11)(13,16,17,19)(14,20,18,15);(5,7)(6,10)(8,11)(9,12);(5,8,9,6)(7,11,12,10)

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307670, #72307711, #72307719
maximal normal subgroup of	#72307670, #72307711, #72307719

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539853);