#7627404

order	512
nilpotency class	2
isoclinism	isoclinic class of #63041
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂²×H(3,𝔽₄)	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7986 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72333436
maximal normal subgroup of	#72333436
maximal quotient of	#72333495, #72334050

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534600);