#7674649

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
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280759
maximal normal subgroup of	#72280759
maximal quotient of	#72281552, #72281666

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7581845);