#7677032

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

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

11900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 16 points:

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

Polycyclic presentation:

2^9|b^2=g,e^2=i,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	#72280980, #72315645, #72315693
maximal normal subgroup of	#72280980, #72315645, #72315693
maximal quotient of	#72281617, #72282296, #72283123

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7584228);