#7659924

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	C₂×#3513	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7135 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)(15,22)(19,26)(23,27)(24,28);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14,18,21)(15,19,22,26)(16,17,20,25)(23,24,27,28);(5,7)(6,9)(8,12)(10,11)(13,15,18,22)(14,19,21,26)(16,23,20,27)(17,24,25,28);(5,6)(7,10)(8,11)(9,12)(13,15)(14,19)(16,23)(17,24)(18,22)(20,27)(21,26)(25,28);(5,6)(7,9)(8,11)(10,12)(13,16)(14,17)(15,23)(18,20)(19,24)(21,25)(22,27)(26,28);(5,7)(6,10)(8,12)(9,11)(13,17)(14,20)(15,24)(16,21)(18,25)(19,27)(22,28)(23,26)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256965, #72257018, #72257123
maximal normal subgroup of	#72256965, #72257018, #72257123

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7567120);