#7631219

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

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

15798 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72228314, #72258719, #72278542
maximal subgroup of	#72225944, #72225958, #72226215, …
maximal normal subgroup of	#72225944, #72225958, #72226215, …
maximal quotient of	#72228377, #72278632, #72299105, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538415);