#7627526

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	C₂×#3558	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7392 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=g,e^2=gi,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	#72238152, #72238165, #72238254
maximal normal subgroup of	#72238152, #72238165, #72238254
maximal quotient of	#72263732, #72263794

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534722);