#7699580

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

8451 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#3481, #72282857, #72282940, …
maximal normal subgroup of	D₄×#3481, #72282857, #72282940, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7606776);