#7695092

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	C₂×D₄²	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7419 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234910, #72237366, #72263970, …
maximal normal subgroup of	#72234910, #72237366, #72263970, …
maximal quotient of	C₂×#7695092

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7602288);