#7687522

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

7755 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)(7,12)(8,10)(9,11)(14,17)(18,19);(1,2)(3,4)(5,6)(7,12)(8,10)(9,11);(5,7,9,10)(6,8,11,12)(13,14,16,17)(15,19,20,18);(5,6)(7,12)(8,10)(9,11)(13,15,16,20)(14,18,17,19);(5,6)(7,8)(9,11)(10,12);(5,8)(6,10)(7,11)(9,12)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=hi,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, #72234924, #72234927
maximal normal subgroup of	#72234910, #72234924, #72234927
maximal quotient of	#72261296, #72261609

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7594718);