#7631968

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

8222 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256602, #72256688, #72256690
maximal normal subgroup of	#72256602, #72256688, #72256690
maximal quotient of	#72256780, #72263474

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539164);