#7792210

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

7645 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281540
maximal normal subgroup of	#72281540

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7699406);