#7919549

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

7688 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72236848, #72238187, #72238225, …
maximal normal subgroup of	#72236848, #72238187, #72238225, …
maximal quotient of	#72336899, #72336935

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7826745);