#7667094

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 20 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315015, #72315195, #72315207
maximal normal subgroup of	#72315015, #72315195, #72315207

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7574290);