#7720362

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

7412 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(2,4)(5,6)(7,8)(9,10)(11,12)(13,14,18,20)(15,22,16,19)(17,25,26,21)(23,27,24,28);(1,2,3,4)(13,15,18,16)(14,19,20,22)(17,23,26,24)(21,27,25,28);(6,10)(7,11)(14,20)(15,16)(21,25)(23,24);(5,7,9,11)(6,8,10,12)(13,16,18,15)(14,19,20,22)(17,24,26,23)(21,27,25,28);(5,8)(6,11)(7,10)(9,12)(13,14)(15,22)(16,19)(17,21)(18,20)(23,28)(24,27)(25,26);(13,17,18,26)(14,21,20,25)(15,23,16,24)(19,27,22,28)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238245, #72238246, #72238255
maximal normal subgroup of	#72238245, #72238246, #72238255

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7627558);