#7857925

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

7268 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)(5,6,10,8)(7,12,9,11)(13,14)(15,20)(16,22)(17,27)(18,24)(19,21)(23,25)(26,28);(1,2)(3,4)(5,7,10,9)(6,11,8,12)(13,15)(14,20)(16,25)(17,26)(18,21)(19,24)(22,23)(27,28);(6,8)(11,12)(14,21)(15,24)(16,26)(23,27);(5,8,10,6)(7,12,9,11)(13,16,19,26)(14,22,21,28)(15,25,24,17)(18,27,20,23);(5,9,10,7)(6,12,8,11)(13,17,19,25)(14,23,21,27)(15,26,24,16)(18,28,20,22);(13,18)(14,24)(15,21)(16,27)(17,22)(19,20)(23,26)(25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238144, #72238154, #72238248, …
maximal normal subgroup of	#72238144, #72238154, #72238248, …
maximal quotient of	#72263446

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7765121);