#7631035

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
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8322 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)(5,6,9,8)(7,11,12,10)(13,14,19,23)(15,24,27,20)(16,17,21,28)(18,26,25,22);(5,7,9,12)(6,10,8,11)(13,15)(14,20)(16,18)(17,22)(19,27)(21,25)(23,24)(26,28);(1,2)(3,4)(5,8,9,6)(7,11,12,10)(13,16,19,21)(14,17,23,28)(15,25,27,18)(20,26,24,22);(13,16)(14,17)(15,18)(19,21)(20,22)(23,28)(24,26)(25,27);(13,17)(14,21)(15,26)(16,23)(18,20)(19,28)(22,27)(24,25);(13,18)(14,22)(15,16)(17,20)(19,25)(21,27)(23,26)(24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313571, #72313631
maximal normal subgroup of	#72313571, #72313631
maximal quotient of	#72316309

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538231);